Tan Theta

The tangent function ‘or’ Tan Theta is one of the three most common trigonometric functions, along with sine and cosine. The tangent function in a right-angle triangle is defined as the ratio of the opposite side to the adjacent side. 

The mathematical denotation of the tangent is,

\(\tan(\theta) = \frac{\text{Opposite Side}}{\text{Adjacent Side}}\)

More About Tan Theta

Tan theta formula can also be calculated from the ratio of the sine of the angle to the cosine of the angle.

\(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)

The derivative of \(\tan(\theta)\) in calculus is \(\sec^2(\theta)\) and the integral of it is \(\ln|\cos(\theta)|\). The reciprocal of tan theta is \(\cot(\theta)\).

Below is a table of tan theta values for different degrees and radians.   

Radians	Degree	Tangent Value
0	0°	0
\(\frac{\pi}{6}\)	30°	\(\frac{\sqrt{3}}{3}\)
\(\frac{\pi}{4}\)	45°	1
\(\frac{\pi}{3}\)	60°	\(\sqrt{3}\)
\(\frac{\pi}{2}\)	90°	\(\infty\)
\(\pi\)	180°	0
\(\frac{3\pi}{2}\)	270°	\(\infty\)
\(2\pi\)	360°	0

Important Tan Theta Formula

Some important properties of the tangent and tan theta formula are:

• \(\tan(-x) = -\tan(x)\)

• \(\tan(90° – x) = \cot(x)\)

• \(\tan(x + \pi) = \tan(x)\)

• \(\tan(\pi – x) = -\tan(x)\)

• \(\tan^2(x) = \sec^2(x)-1\)

• \(tan(x + y) = \frac{\tan(x) + \tan(y)}{1 – \tan(x)*\tan(y)}\)

• \(tan(x – y) = \frac{\tan(x) – \tan(y)}{1 + \tan(x)*\tan(y)}\)

• \(\tan(2x) = \frac{2 \tan(x)}{1 – \tan^2(x)}\)

• \(\tan(3x) = \frac{3 \tan(x) – tan^3(x)}{1 – 3 \tan^2(x)}\)

• \(\tan(\frac{x}{2}) = \sqrt{\frac{1 – \cos(x)}{1 + \cos(x)}} = \frac{1 – \cos(x)}{\sin(x)} = \frac{\sin(x)}{1 + \cos(x)}\)

Solved Examples

Question 1. If \(\sin(x) = \frac{4}{5}\), calculate the value of \(\tan(x)\).

Solution. Using trigonometric identity, 

\(\cos^2(x) = 1 – \sin^2(x) = 1 – \frac{16}{25}\)

\(\cos^2(x) = \frac{9}{25}\)

\(\cos(x) = \frac{3}{5}\)

Now,
\(\tan(x) = \frac{\sin(x)}{\cos(x)} = \frac{4/5}{3/5}\)

\(∴ \tan(x) = \frac{4}{3}\)

Question 2. If \(\sec(x) = \frac{5}{8}\), calculate the value of \(\tan(x)\).

Solution. Using trigonometric identity, 

\(
\begin{align}
\tan^2(x) & = \sec^2(x) – 1\\
& = \frac{5}{8}^2 – 1\\
& = \frac{25}{64} – 1\\
& = -\frac{39}{64}\\
\tan(x) & = \sqrt{-\frac{39}{64}}\\
\end{align}
\)

Question 3. John is standing on the ground and looking at the top of a tower with an angle of elevation of 60°. The distance between John and the tower is 15 feet. Calculate the height of the tower.

Solution. As we know,

\(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}\)

\(\tan(60°) = \frac{\text{Opposite}}{\text{15}}\)

\(\sqrt{3} = \frac{\text{Opposite}}{\text{15}}\)

\(\text{Opposite} = 15 \sqrt{3}\)

∴ The height of the tower is \(15 \sqrt{3}\) feet.

FAQs

More Trigonometric Function

Sin Theta	Cos Theta
Sec Theta	Cot Theta
Cosec Theta